Now, the non-equivalence is a problem for biological classifications in general; thus there is continual debate as to whether "birds" should be classed as "reptiles" or "aves". But on the individual level 1-6 tightly correlate.

Quote:Real genetic similarity cant be.. What counts is expression of the gene and thus phenotype especially in complex traits and in this case people from different races are frequently more alike to one another than to members of their own race and also in many cases by very large margins.

If you are going to comment here, do not employ slushy phrases like "real genetic similarity". What counts as "real" for you may not be the same as what counts so for e.g., a proponent of Salterian ethnic genetic interest (who would actually care about the total number of shared alleles).

That said, you seem to be just restarting Lewontin's fallacy. In this or that particular trait "people from different races are..." but not on a multivariate measure. Del Giudice, et al. (2012) had a nice paper on gender differences which illustrates the same point, "The distance between Mars and Venus: Measuring global sex differences":

[/url]

In a follow up, Del Giudice cogently rebuts Lewontin-like replies which argue that an average univariate approach gives a truer picture of reality than a multivariate one:

Quote:The argument is thus revealed as a non sequitur: clearly, it makes perfect sense to speak of large sex differences in facial morphology—“lumping together” differences in multiple traits such as face width and eye size—with no need to assume that individual differences in facial traits can be reduced to a single dimension (see Figure 1). Far from being uninterpretable, the resulting axis of individual variation can be easily defined as facial masculinity-femininity (e.g., Hennessy et al., 2005) or some equivalent term. As is apparent from Figure 2, facial masculinity-femininity is a very recognizable configural trait that summarizes a multitude of individual morphological differences, even if it does not correspond to any specific anatomical structure. In exactly the same way, multivariate sex differences in personality can be interpreted as defining an axis of individual variation in personality masculinity-femininity (see Lippa, 2001). Whatever the exact terms chosen to denote these constructs, there is nothing mysterious about their interpretation. Of course, the global differences measured by D are not intended to replace univariate effects but rather supplement them, as stressed in Del Giudice (2009).

Zoologists and biologists do recognise the convenience of many non-human races, even when they only capture a small amount of variation. However the small amount of variation is always structured so this variation is useful:

"It is true that some biologists have seen fit to create new [sub]races among lower animals on the basis of such single slight characters as differences in pigmentation of the hair on a part of the tail. Such a procedure would be perfectly justifiable if it were taxonomically helpful. Nor would it be necessary to stipulate that animals in other groups shall not exhibit this character, but one would have to insist that almost all members of one or both sexes of the new [sub]race shall exhibit it. No such requirement is fulfilled by the races and subraces that Coon created." (Montagu, 1997: 50-51)

When it comes to human races this is simply not the case.

So race must capture either:

(a) A lot of variation.
(b) A small amount of variation, but which is structured in such a way it is still useful to classify.

Strongly disagree with how you interpret Brace et al (1993). The paper repeatedly clarifies "clusters" are not races. C. Loring Brace understands that what he describes as non-adaptive craniometric variation plots by geographic distance and so the "clusters" (as groupings of populations) are arbitrary:

"Figure 2 might be construed as providing support for the hoary folk belief that modern Homo sapiens can be sorted into three convenient “races”: “caucasoid,” “mongoloid,” and “negroid.” When the number of separate regional representatives is multiplied, however, it becomes clear that the ties between adjacent twigs on the dendrograms are simply indications of the extent to which geographically adjacent people are genetically related to each other rather than the extent to which they reflect anything that could be called a “racial” essence."

And in another paper:

"The farther away from the center of a geographic region one goes, the less the peripheral members of that cluster share with the others in an opposite direction and the more they share with members of the immediately adjacent geographic region." (Brace, 2000)

This is the opposite of traditional races, which proposed there was abrupt change or sharp discontinuity between human populations.

The argument then becomes are these arbitrary groupings or clusters convenient?

Brace doesn't seem to think so: he calls them "trivial" and "unimportant".

Why then did he publish a study using these arbitrary clusters if he doesn't find them useful? The study was on the ancient Egyptian race controversy, so it was published in response to the public and media that is obsessed with race.

(2015-Jul-27, 02:09:46)Krom Wrote: This is the opposite of traditional races, which proposed there was abrupt change or sharp discontinuity between human populations.

As held by whom? Buffon, Blumenbach, Esper, Prichard, Darwin, Quatrefages ...? Specifically, list the many early (hence "traditional") proponents of race (as divisions of a species) who maintained that these divisions were -- moreover were definitionally so -- marked by "abrupt change" or "sharp discontinuity". I do not want to hear Naomi Zack's bullshit; I want specific "traditional" references. I had, in my paper, noted:

"More generally, one wonders when biological races, in the intraspecific sense, were in general considered to be sharply discontinuous. Did Dobzhansky and Boyd conceptualize them as such? Did Mayr conceptualize his microgeographical races so? Did Blumenbach, Buffon, or Thomas Huxley with his Melanochroi that “pass by innumerable graduations into the Australoid type of the Dkhab, while in Europe they shade off by endless varieties of intermixture into the Xanthochroi (1870)”? Nope. What about the many early 20th century anthropologists who equated ethnic groups with races? Not generally. What about races understood as the “mere” or “accidental” varieties of the 18th and 19th century (see: Stamos (2012))?"

To this list I can now add dozens of names. We seem to not have made it beyond the point in bold:

If you read Huxley (1870) properly you will see he says there are traits that have no variation inside races, e.g. "Australians are invariably dolichocephalic", Negroid hair is "always short and crisp or woolly", and Bushmen "are all dolichocephalic", etc.

All these claims are false, for example Larnach and Macintosh (1974) found 17.7% of aboriginal crania from the Cairns Rain Forest and 9.6% from Queensland to be non-dolichocephalic (mesocephalic).

The sources (like Huxley) you quote don't support what you are saying. I can't be bothered to go through all of them, but here's Boyd (1958):

"I have previously suggested a definition of a human race which I
propose to use here: "a population which differs significantly from other human populations in regard to the frequency of one or
more of the genes it possesses."

"There are clear-cut differences in blood group gene frequencies
which distinguish a number of human populations one from the
other, and we may accordingly call these populations races."

Dobzhansky also wrote: "The sharper the discontinuities, the more
"‘natural’’ the races’ (1950 [1946], p. 120). See Gannett (2013).

"differs significantly", "clear-cut", "sharper the discontinuities"... could it be any more clearer?

The fact is you cannot show this and are therefore forced to defend a trivial concept of race that has little in common with how races were traditionally defined.

"Race naturalists are acknowledging that strong racial naturalism is not supported by the relevant science, and they are falling back on weak definitions of race that need to be defended on semantic grounds. For instance Sesardic writes, “In principle we might introduce names for hundreds or even thousands of human groups that we could call races on the grounds of their genetic differentiation” (Sesardic, 2013, p. 290). This is a very weak form of racial naturalism, of the race-as-population variety." (Hochman, 2014)

for example Larnach and Macintosh (1974) found 17.7% of aboriginal crania from the Cairns Rain Forest and 9.6% from Queensland to be non-dolichocephalic (mesocephalic).

Krom,

Three points:

1. By the same line of reasoning, one could show that species don't exist. Our own species varies considerably in dental and cranial traits, and this variation overlaps not only with what we see in Neandertals but also with what we see in some nonhuman primates. The same with blood groups. The same with a lot of other traits.

2. I don't want to question your intellectual honesty, but please take a minute to read what you write before you publish. You quote several authors to show that the race concept implies "clear-cut" and "sharp discontinuities" between human races. In fact, the actual quotes refer to "clear-cut" differences in trait frequencies and "sharper discontinuities" (which is admittedly an oxymoron). In both cases, the authors defined human races as fuzzy sets. If human races were indeed sharply defined entities, they would not be races. They would be species. Indeed, even species are not as sharply defined as we like to think.

3. Quoting Montagu to defend your position on race is like quoting Margaret Thatcher to defend strike breaking. As a general rule, it's disreputable to defend one's opinion by simply quoting another person's opinion. This is the fallacy of "appeal to authority." It's even more disreputable if the other person is neither a neutral observer nor an eminence grise. Ashley Montagu was neither.

(2015-Jul-27, 17:56:05)Krom Wrote: If you read Huxley (1870) properly you will see he says there are traits that have no variation inside races, e.g. "Australians are invariably dolichocephalic", Negroid hair is "always short and crisp or woolly", and Bushmen "are all dolichocephalic", etc.

You are, of course, confusing issues, a strategy which you seem to frequently employ. Your claim above was that the "traditional" concept of race was one of deeply discontinuous divisions of a species. Since there have been multiple formulations of the race concept, the most obvious interpretation of this claim is: between the 18th and early 20th century races [i]as such were typically thought of as deeply discontinuous divisions.[/i] Now to be clear, since you seem to not be the most philosophically inclined individual, the claim, for your argument to work, must be that the concept of race, not merely the understanding of specific races, was one of deeply discontinuous divisions. Michael Hardimon has well clarified this distinction between definitions or concepts of race and perceptions of specific classifications. To give a concrete example, Coon's major races were (prior to the proposed time of admixture) deeply discontinuous, since he adopted a multiregional model. However, since he allowed for numerous "local races" which were not deeply discontinuous, one can not say that his concept of race entailed deep discontinuity. Given the aforesaid, as counter evidence to your claim, I need only provide the following types of evidence: (1) claims made during the said time that races as such (that is, definitionally) did not entail deep discontinuity, (2) discussions of races which the authors acknowledge were NOT deeply discontinuous, or (3) discussions which imply (1) or (2). Now, the same can be said in regards to your claim that traditional race concepts were ones of purely homogeneous groups. Before proceeding, let us be clear regarding what we mean by "deep discontinuity". We do not mean merely discontinuous, let alone non-overlapping. Adjacent non-overlapping divisions and ones with minor discontinuous are, to be sure, distinct, where distinct means recognizably different but the intervals between divisions would not be large, where "large" is somewhat subjectively defined. Your claim is that divisions were thought necessarily to have large gaps between them, as species were thought to.

Now as I have noted, this assertions of your makes no sense. Even those who argued that races were originally created discontinuously, that is, represented separate creations, granted the admixture between them, the blending of races; they merely argued that hybridization did not constitute evidence against separate origins. Unlike you, they still recognized these admixed and blended groups as being races, just admixed ones. But I suppose that you will only be satisfied if I provide a plethora of references:

1. Buffon (1781): "The blood of the Tartars is mixed on one side with the Chinese, and, on the other, with the oriental Russians. But the characteristic features of the race are not entirely obliterated by this mixture; for, among the Muscovites, the Tartarian aspect is very frequent; and, though the former have sprung from the common European race, we still find many individuals with squat bodies, thick thighs, and short legs, like the Tartars."

(Buffon's mixed Tartars obviously are not deeply discontinuous with respect to Chinese; the same author situated his races of dogs in a genealogical network; these races again were not in any sense deeply discontinuous.)

2. Zimmerman (1783): "The opponents of the <climatic explanation of the origin of the Negro> are used to citing the thick lips, indented nose, and especially the wool or woolied hair of the Negro <in support of> their double lineal stem stock. I confess, however, that for me these three <features> are of no special importance. For thick lips can be found everywhere. The Eskimo and Kalmuck also have them, and we can easily adduce <examples of> thick-mouth families. Besides, there are Negroid nations that have neither thick nor indented lips... Second, the division of human-kind <into distinctly different groups> is still difficult in consequence of an important collateral cause, namely, the migration of peoples and the interbreeding that arises as a consequence of this."

[Zimmerman recognized that population flows made distinctly different divisions difficult.]

3. Blumenbach (1806): Neither must we take merely one pair of the races of man which stand strikingly in opposition to each other, and put these against the other, omitting all the intermediate races, which make up the connection between them. We must never forget that there is not a single one of the bodily differences in any one variety of man, which does not run into some of the others by such endless shades of all sorts, that the naturalist or physiologist has yet to be born, who can with any grounds of certainty attempt to lay down any fixed bounds between these shades, and consequently between their two extremes.

(This is a very clear statement on the matter.)

4. Lamarck (1809): "If all the races (so-called species) belonging to a kingdom of living bodies were thoroughly known, as well as their true affinities, so that the sorting out of these races and their allocation in various groups were in conformity with their natural affinities... On such an assumption, nothing doubtless would be more difficult than to assign the boundaries between these different divisions; arbitrary opinion would produce incessant variation, and there would be no agreement except where gaps in the series made clear demarcations... Fortunately for the practicability of the artifices which we have to introduce into our classifications, there are many races of animals and plants that are still unknown to us."

(As an evolutionist, Lamarck did not draw a strict distinction between races and species. He seemed to appreciate the general distinction, though, as he noted: "However, this means alone suffices to gradually create the varieties which have afterwards arisen from races, and which, with time, constitute that which we call species.")

5. Prichard (1855): "The different races of men are not distinguished from
each other by strongly marked, uniform, and permanent distinctions, as are the several species belonging to any given tribe of animals. All the diversities which exist are variable, and pass into each other by insensible gradations; and there is, moreover, scarcely an instance in which an actual transition cannot be proved to have taken place."

(As a dogmatic monogenist Prichard emphasized the inconstancy of races.)

6. Quatrefages (1861): "Even if he would be so, when even these differences are also insignificant [such that one] wants to say: who cares? From the moment they became constant and they transmit by way of inheritance, they constitute no less real races.... Then for distinguishing species of races, we compare as many samples as possible. Whenever between two forms, even very different, one can establish a graduated series of individuals passing from one to the other by insensible shades, whenever you see above characters intercrossed in terms of this series, we can ensure that both forms belong to a same species. Indeed, between two species even extremely close, there has never exchange or mix of specific characters [between] them."

[The translation is not perfect (the original being in latin), but the general idea is that races, unlike species, interbreed and, as such, graduate into one another by insensible shades.]

7. Darwin (1871): But the most weighty of all the arguments against treating the races of man as distinct species, is that they graduate into each other, independently in many cases, as far as we can judge, of their having intercrossed...

(Here Darwin uses "race" is the inclusive sense which includes both constant varieties and species; as an evolutionist he did not draw a strict distinction.)

8. Huxley (1870): Melanochroi that “pass by innumerable graduations into the Australoid type of the Dkhab, while in Europe they shade off by endless varieties of intermixture into the Xanthochroi (1870)”.

(I stand by my citation; if Melanochroi pass into Australoid, they can not be "deeply discontinuous"; as for Huxley (1870) statement that: "The hair is usually raven-black, fine silky in texture; and it is never wooly, but usually wavy and tolerably long...The Australians are invariably dolichocephalism, the cranial index rarely exceeding 75 or 76, and often not amounting to more than 71 or 72", your are reading too far into "invariably"; dolichocephalism is defined as a cranial index not exceeding 75; "rarely exceeding" implies "some exceeding", thus "invariably" means "almost always". Now, what's notable is Huxley's continual qualification "usually", "often", "sometimes", "generally", "varies", "commonly". He clearly recognized individual variation, which disconfirms your claim that primary races/types were seen an uniform. Whether or not he overestimated the similarity between individuals of certain race in specific characters is irrelevant, since he obviously is not defining race membership in terms of character similarity; thus his dolichocephalic Xanthrocoid remain members of a separate race. Recall again that the concern is with race concepts, not race classifications.)

Now, I could go on. And were I to I would find no evidence in defense of your proposition. If you have it, though, please provide it. That is, I would like specific references which indicate that races were defined or conceptualized as deeply discontinuous divisions. Note that locating these is a more difficult task that what I engaged in above. For while it is sufficient for me to show that a particular author's races were not seen as deeply discontinuous to establish that that author's concept did not necessitate such discontinuity, the reverse does not hold. Showing that some of a particular author's classifications were deeply discontinuous, does not establish that his or her concept entailed this. This should be an obvious logical point, but one you seem not to well grasp, given your tendency to confuse concepts with classifications.

But you said:

Quote:The sources (like Huxley) you quote don't support what you are saying

I quoted Huxley in regards to your claim that race was traditionally understood to refer to sharply discontinuous divisions (a claim, which I again emphasize is not equivalent to that that certain races were seen as sharply discontinuous. For surely one can maintain that Ostrich races are sharply discontinuous, without maintaining that all races need to be so. Huxley's essay also happened to support my contention that races as types were not seen as lacking individual variation. In regards to your point, it doesn't go very far. So he expressed the view that e.g., Austroads never have wooly hair -- even if we take such statements literally, that only tells us that he made an empirical mis-estimation.

You continue:

Quote:I can't be bothered to go through all of them, but here's Boyd (1958):

"I have previously suggested a definition of a human race which I
propose to use here: "a population which differs significantly from other human populations in regard to the frequency of one or
more of the genes it possesses....There are clear-cut differences in blood group gene frequencies which distinguish a number of human populations one from the other, and we may accordingly call these populations races."

The excerpts below summarize Boyd's book:

Boyd (195). Genes and the Races of man:

Quote:In classifying plants, botanists found that if a "natural" classification is desired, that is, one which gives information as to degree of relationship of the various species, it is best not to use vegetative characters...This principle may also apply in some cases to attempt to recognize races within a species... A contemporary physical anthropologist, Dr. William Howells, implies in his recent book, Mankind So Far, that apart from making a judicious guess, he can not always identify race by measurements of any single skull, and the rest of the skeleton gives almost no indication at all...As Dobzhansky and Epling point out, " a system of morphological averages may well serve as an exploratory device" (in taxonomy) but these workers go on to warn us that a basic understanding of the principles of racial variation can come only from knowledge of the distribution of the relative frequencies of variable genes and chromosome structures in a populations. The difficulties in the above instance are party due to the underlying assumption that any individual of one race should be distinguishable from any individual of any other race. This is not generally true, and we shall see that a more modern concept of race does not assume it... It is the thesis of this book that racial categories, if they are to have a valid conceptual basis, must be made on the basis of man's genetic constitution...How is race to be defined? What are the differences, if any, which will cause us to arrange human beings into several distinct categories?... So far as common descent goes, this can be inferred, but seldom proved. Even when geographical differentiation is observed in the species, this does not definitively prove common descent for the population in the area in question... Therefore, there is no necessary justification in assuming that a group of people we consider sufficiently alike to constitute a race do actually derive from common origin, unless we can prove it by historical or paleontological evidence... Less exacting definitions of a "pure race" have been offered; Scheidt simply requires that the several contributing elements have become so completely blended that correlations fails to reveal their original combination... [t]his definition is unobjectionable, although probably not very useful in practice.... All anthropologists understand, but some laymen apparently do not, that no one physical characteristic is going to be enough to enable us to distinguish the various races of man, one from another....Classifications based on the type of hair, whether straight, wavy, or kinky, etc. are perhaps satisfactory as any classification based on a single characteristic can be. But as soon as the whole population of the world is included in this scheme strange contradictions and inconsistencies begin to appear... Races, on the other hand, are more or less genetically open systems. Their populations are channels through which genes can and do flow from race to race.... The two [species] clusters are entirely separate because there are no intermediates, and we can say with perfect safety that any possible cat is different from any possible lion, and that cats as a group are different from lions as a group...[races are not like this]... Since the genes may vary independently one from the other, an individual may carry some genes which occur more frequently in the representatives of one population or race and other genes which are more characteristic of another. Such an individual could be said to belong to two or more races (depending on our definition of race) at the same time, since he is in fact compounded of elements of both... Taken in conjunction with the fact that individual members of the same race, e.g., the white race, or the black race, as a rule differ from each other by many more genes than this, the forgoing estimate makes it seem quite probable that the intrinsic differences between the various races are not large or important in the genetical sense. The genetic differences between different individuals of the same race can often, even with out present limited knowledge of human genetics, be demonstrated to be as large as ten or twenty genes. Considering the enormous number of heredity differences still unanalyzed, it is likely that two humans beings can differ in respect to a thousand or more genes, without seeming to use to be different enough to put into different racial categories...Within this species there are various groups which differ from each other to a greater or lesser degree and some of these we may if we choose, define as races... A race is not an individual and it is not a single genotype, but it is a groups of individuals, more or less from the same geographic area, usually with a number of identical genes, but in which many different types may occur... In the ideal case, one would take account of all the variable genes and chromosome structures in order to describe a given race... Because of the incompleteness of our knowledge, we have to base our classification on certain genetic differences which we do know about... We still have to decide how much difference there must be in the distribution of the known variable genes and chromosome structures between two populations before we decide to call them two different races... Statistically significant differences may occur between populations of Drosophila in localities only a dozen miles apart. Therefore we could if we liked say that these populations are racially distinct. If the concept of race is to be taxonomically useful, however, we should not use the term quite s freely, for otherwise we shall have too many races within each species, and the term will lose much of its value for classification... To make the race concept useful, we must look for some additional feature to use in defining it....when this is true we have a geographical gradient (cline), analogous, to the slopes shown by the contour lines on topographical maps, or the temperature gradient crossing the isotherm on weather maps... Populations at two ends of such a gradient may be profoundly different in genetic constitution, but they may be connected by all grades of genetic variation between the two. In such a case, whether or not the systematist decides to break up the population into two or more sections and designate them as races is quite arbitrary. This decision will be based on considerations of expediency and nothing else. In favor of defining two or more races there will be the advantage of having unique simple reference names which apply to the various populations... Dobzhansky and Epling note that the gene frequency gradient between some populations may be relatively steep, and that in some cases these gradients are steep enough (contour lines close together) to justify taking some point of the steep portion as a boundary and thus breaking up the species into one or more discrete arrays of populations which will have few intermediates. In such a situation we have little hesitation in defining races. Each major population array becomes a race, and these arrays can be delimited, counted, and named. The mean differences between these population arrays may not necessarily be any greater than those between the end members of a continuous population chain which we may prefer to call one (variable) race, but if nature, by providing us with complete or nearly complete discontinuities between arrays, has to a large extent eliminated the arbitrary human element otherwise involved in drawing dividing lines between populations, we feel justified in referring to these array (isolates) as races... In nature the geographic gradients in gene frequencies are seldom quite uniform on the one hand and are seldom entirely discontinuous on the other, so we have usually to deal with a somewhat intermediate situation. Therefore, it is not all uncommon to find within a single species that two certain races may be easily separated from each other, while two other races show only a slight discontinuity where there populations meet or overlap. Whatever the course adopted in devising names for the populations, the realities of the situation are not altered in the least. It still remains a fact that the species Drosophila pseudobscura is geographically differentiated, and that populations of different territories differ in regard to the relative frequencies of chromosome types (and genes). And this is exactly what the geneticist means by racial differentiation... [W]e must not expect that the various data collected will all be mutually "consistent". In fact, we must be prepared to find the populations in Greenland and in Australia agree quite well in regards to blood groups frequencies (A, B, O at least), while they markedly disagree in regards to the M and N blood types, in regards to skin color, hair form, and other characteristics.... It is to be feared that many anthropologist, and many laymen nevertheless have continued to hope that human races are so different basically that differences will show up, almost irrespective of what sort of observations we make on populations. This attitude interposes an unfortunate obstacle to progress towards a really objective concept of human races.... As data on the physical characteristics of the human race have accumulated, however, and it has become clear how they are mostly independent of each other, this point of view has become more and more patently absurd, and it has gradually become clear that whatever races we choose to distinguish will be almost entirely arbitrary, and their distribution will depend on the particular characteristics on which we choose to base them...We may define a human race as a population which differs significantly from other human populations in regards to the frequency of one or more of the genes it possesses. It is an arbitrary matter which, and how many, gene loci we choose to consider as a significant "constellation"; but it seems better on the one hand, not to designate a multiplicity of races which differ only in regards to a single pair or a single set of allelic genes; and on the other, not to insist that the races we define must differ from each other with respect to all their genes....Since every human individual has to belong to one or another of the four blood groups, the only racial difference we find, or can expect to find, will be differences in frequencies of the four groups among various populations... The American Indian, for instance, known to be derived by direct descent largely or at least partly from Mongoloid ancestors emigrating from the Asiatic mainland, would have to be placed in a quite different race on the basis of blood groups. This only serve to emphasize the meaning which is being given to the term "race" in this book...The very idea of racial differentiation implies that geographically isolated groups, although ultimately of the same origin may eventually come to differ... It must not be thought that the divisions between our genetic races will be absolutely sharp, any more than is the difference between races which are characterized by any other method. Isolation has not been enough for that... It will be noted that our proposed racial classifications, although it is based upon gene frequencies, as we decided a valid classification must be, does not really differ in any very startling way, in so far as the ultimate categories are concerned, from some of the older classifications...Consequently [previous classifiers] always reached...a final goal of human races distributed roughly according to geography and common descent. All we have done is to show that the same thing can be accomplished more simply, and without so many inconsistencies in the application of our method, by considering genes frequencies.

There are obvious points on which I disagree with Boyd and the other early "population" geneticists. Boyd for example notes:

Quote:The difficulties in the above instance are party due to the underlying assumption that any individual of one race should be distinguishable from any individual of any other race. This is not generally true, and we shall see that a more modern concept of race does not assume it...

This data on the physical characteristics of the human race have accumulated, however, and it has become clear how they are mostly independent of each other, this point of view has become more and more patently absurd, and it has gradually become clear that whatever races we choose to distinguish will be almost entirely arbitrary

While they granted that racial divisions should be based on as many genes as possible, they didn't realized -- or, if you believe Hooton, they didn't want to realize -- that one could use numerous molecular character to unequivocally classify individuals into "populations" and that the vast majority of alleles correlated allowing one to make non-arbitrary divisions (e.g., using cluster analysis). As a result, their "populations" were conceptually different from the modern population geneticist's genomic population, which represent cluster classes. (Read: The concept of "population", with respect to race, has shifted back to a class based one -- this is a point which you have been missing. The new population concepts are class ones such as, "In population genetics, a race is a group of organisms in a species that are genetically more similar to each other than they are to the members of other such groups". This noted, your point is not supported. Boyd clearly did not require deeply discontinuous populations nor did he think that populations differences involve more than frequency ones.

But you say:

Quote:Dobzhansky also wrote: "The sharper the discontinuities, the more
"‘natural’’ the races’ (1950 [1946], p. 120). See Gannett (2013).

Meaning, as Boyd noted, the less arbitrary, that is, the more nature cuts out the partitions for us via providing barriers to reproduction. I noted the same in my discussion of "arbitrary" versus objective delineations (section II). Nonetheless, the not so sharp divisions are still races. Indeed, Dobzhansky makes the common point that one can cut races from a continuum. And discussing minor races, he notes: "One may perhaps question the desirability of applying the term ‘racial differences’ to distinctions as small as those that can be found between populations of neighboring villages and as large as those between populations of different continents. Might one modify the definition of race by specifying that the differences in gene frequencies be above a certain minimum magnitude? Such a modification is undesirable for two reasons..." "Sharp discontinuities" is hardly a part of his definition of race!

Most, if not all, your sources are saying gradients ("intermediate" populations) are the product of race mixture/migration, not in situ. This means these 18th-19th century scientists thought there are/were homogenous races, and the continuum was the product of these sharply discontinuous races. You can't have it any other way.
This confirms what I have already posted, but here's an extract from Grover Krantz's Climatic Races and Descent Groups (1980):

"Many people see themselves divided into two hostile camps on this subject, the "clinists" vs. the "racers." (I hesitate to say "racists" because that implies more than is usually the case.). The basic race idea, as discussed above, is that races are natural units that tend to maintain their identity. When physically intermediate populations are noted, the racer will say they are mixtures or hybrids of the more "real" races. This has sometimes been called anti-evolutionary because of the implied permanence of each race. Those who adhere to the basic race concept usually do not make it clear just why the races should be considered to be so distinct. This is often treated as simply self-evident."

He then discusses the opposing "clinalist" (non-racial) view:

"A cline is a slope, incline or decline. In this case it refers to the slope on a graph which illustrates how the degree of expression of a physical trait changes over space. A simple example of this would be a graph of the average amount of melanin (dark coloring) found in human skins along a north-south line from Poland to Nigeria (See Fig. 1.). There is little melanin in skins at the north end of this line and it gradually increases to a maximum amount at the south end. The line of melanin quantity thus slopes; it is a cline. A clinist then says that's all there is to race; just populations located at different positions along a continuum, or cline, of a particular trait."

Lastly he clarifies that "race" requires a steepness in a cline: "the steep part of the cline is called a racial boundary by the racers."

Of course "clinists" do not regard there to be steep clines (sharp discontinuity) that would justify the classification of races. If these were shown we all would be "racers".

Quote: Now, I could go on. And were I to I would find no evidence in defense of your proposition. If you have it, though, please provide it. That is, I would like specific references which indicate that races were defined or conceptualized as deeply discontinuous divisions.

Your sources say there are races and "intermediate" variations between these races are the product of interbreeding. This means each of the races are deeply discontinuous divisions. I don't need to provide sources when all your sources support what I have posted.

"While events in the Iberian peninsula may have provided the initial stirrings of proto-racial sentiments, the philosophical concept of race did not actually emerge in its present form until the 1684 publication of “A New Division of the Earth” by Francois Bernier (1625–1688) (Bernasconi and Lott 2000, viii; Hannaford 1996, 191, 203). Based on his travels through Egypt, India, and Persia, this essay presented a division of humanity into “four or five species or races of men in particular whose difference is so remarkable that it may be properly made use of as the foundation for a new division of the earth” (Bernasconi and Lott 2000, 1–2)."

"Kant argued that all humans descend from a common human “lineal root genus” in Europe, which contained the biological “seeds” and “dispositions” that can generate the distinct physical traits of race when triggered by divergent environmental factors, especially combinations of heat and humidity. This, combined with patterns of migration, geographic isolation, and in-breeding, led to the differentiation of four distinct, pure races: the “noble blond” of northern Europe; the “copper red” of America (and east Asia); the “black” of Senegambia in Africa; and the “olive-yellow” of Asian-India. Once these discrete racial groups were developed over many generations, further climatic changes will not alter racial phenotypes (Bernasconi and Lott 2000, 8–22)."

"Such inter-racial mixtures accounted for the existence of liminal individuals, whose physical traits seem to lie between the discrete boundaries of one of the four races; peoples who do not fit neatly into one or another race are explained away as groups whose seeds have not been fully triggered by the appropriate environmental stimuli (Bernasconi and Lott 2000, 11)."

^ Especially read that last paragraph. This is what I've repeatedly tried to explain to you.